Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 223 / 3

Let $\Delta ={\theta }_1-{\theta }_0>0$. For one observation from the unit-variance normal location family,The sample log-likelihood ratio is therefore $n{T}_n$. By the Neyman-Pearson lemma, the level-$\alpha$ most powerful test rejects for a sufficiently large likelihood ratio, equivalently when $T_n>C_{\alpha }$, with $C_{\alpha }$ chosen to give null rejection probability $\alpha$.

The functional isThe influence function of an expectation functional isBecause $\psi$ is affine with nonzero slope, this is unbounded under both $F_{{\theta }_0}$ and $F_{{\theta }_1}$.

Now letThenSince $a\le {\psi }_{a,b}(x)\le b$, this influence function is bounded for both hypotheses. Clipping the log-likelihood contribution prevents one extreme observation from having unbounded effect on the statistic.

Put $r(x)=f_{{\theta }_1}(x)/f_{{\theta }_0}(x)$. The proposed first density can be writtenIts integral is continuous and strictly decreasing in $c$, tends to infinity as $c\downarrow 0$, and tends to $1-ϵ$ as $c\to \infty$. Hence a unique $c>0$ makes its integral one. Since $g_{0,c}\ge (1-ϵ)f_{{\theta }_0}$,for the density $h_0=[g_{0,c}-(1-ϵ)f_{{\theta }_0}]/ϵ$.

Similarly,Its integral is continuous and strictly increasing from $1-ϵ$ to infinity as $d$ ranges from zero to infinity. The unique normalizing $d>0$ givesfor a density $h_1$. Thus $G_0\in {\mathcal{P}}_{ϵ}(F_{{\theta }_0})$ and $G_1\in {\mathcal{P}}_{ϵ}(F_{{\theta }_1})$.

Assume $d<c$. Directly comparing the two piecewise densities givesSincewe obtainThus the sample log-likelihood ratio is $n{S}_n$ with truncation values $a=\log d$ and $b=\log c$. Rejecting for large $S_n$ is exactly the likelihood-ratio test between the two least-favorable contaminated distributions.